The present invention is generally directed to fluidic components. A fluidic component in its broadest sense, may be considered a channel that permits flow of fluid therethrough. Non-limiting examples of fluidic components include fluidic mixers, reactors, straws, tubes, etc.
Computational fluid dynamics (CFD) is playing an increasing role in the development of fluidic components, but the automatic design and optimization of new components using CFD is the exception and not the rule. Computational studies of the flow fields within components are more often used to fine-tune a near-final design or provide detailed analysis once a design is finalized. When CFD is used in design, it generally serves to replace some benchtop experiments with numerical experiments, but it leaves the “build-and-test” approach intact. Intuition and experience guide the designer through the arduous process of testing and altering a design until a satisfactory component is produced.
One approach to optimizing a predetermined design starts by defining the geometry using a set of control points, and then moving these points to deform the geometry in an attempt to improve performance. This approach is rare in fluidic applications due to the computational intensity of optimizing complex structures. Some have pursued general shape optimization for microchip electrophoresis. Starting with a bend with consistant inner and outer radii, the shape of the inner wall was adjusted to minimize sample dispersion around the turn. The transport was modeled as two-dimensional electrokinetic flow, and the evolving geometry was modeled in one of two ways: one approach used two parameters to define the shape of the inner wall, and a second discretized the wall using 200 elements. Later work included additional designs that adjusted the shape of the outer wall as well.
A more common approach to design optimization in fluidics is to take an established component and conduct parameter studies using CFD modeling of either the governing flow equations or some reduced, representative model. For example, several studies have examined the effect of varying certain geometric factors in an established herringbone mixer. The complex three-dimensional flow field in a specific herringbone mixer has been approximated using a two-dimensional lid-driven cavity model that was tuned to provide qualitative agreement to experimental data. The effect of varying two geometric parameters was studied: the offset (i.e., the location of the “elbow”) in the groove and the number of grooves in each cycle. Others presented CFD results for six mixers, based on the herringbone mixer design obtained by varying the depth and width of the grooves and the number of grooves per cycle. Still others have studied the effects of varying herringbone offset, depth, and angle, as well as the inflow geometry, by applying CFD to nine configurations chosen judiciously to isolate the effects of the various parameters. These studies propose guidelines for improving the initial herringbone design within the envelope defined by the simulations, but they apply only to mixers that repeat the same feature with a periodic change in orientation.
Other authors pursue component-based optimization in which predefined elements are combined to produce a more complicated system, the entirety of which is then studied using conventional CFD equations. Some describe a component-based approach for modeling electrokinetic flow networks. They use analytical models to predict cross-channel diffusion and streamwise dispersion for two elements: straight channels and 90° bends. Models for tapered channels and T-intersections were introduced in later work. Combining these elements gives composite geometries, and the transport of species through the system is predicted by combining the effects predicted by the analytical models. Others have applied these analysis tools for designing separation systems. In addition to straight sections and bends, they include elements for injectors and detectors, and they use heuristic rules for element placement to generate composite systems automatically, which occupy minimal areas. The analytical models used to characterize element performance do not include vertical variation within the channels.
A key parameter for characterizing viscous flow is the Reynolds number, Re, defined asRe=ρvL/μ  (1)where ρ is the density, v is the characteristic velocity, L is the length scale and μ is the viscosity. The Reynolds number measures the relative importance of inertial effects to viscous effects. Some fluidic systems operate at low Reynolds numbers, which also means that the fluid transport through the components is laminar. For some aqueous fluidic systems, Re is often sufficiently low that the inertial terms in the Navier-Stokes momentum equation are negligible compared with the viscous and pressure-gradient terms. Therefore, the Stokes equations can be used to characterize the flow. Assuming Stokes flow also ensures that a component will perform as designed if further miniaturization is pursued. A mapping strategy in accordance with an exemplary embodiment of the present invention and described below does not require this assumption, however, and the implications of including the inertial terms are discussed further below.
An analyte i present in the fluid moves through a component due to advection and diffusion,
                                                        ∂                              ρ                i                                                    ∂              t                                =                                                    -                u                            ·                              ∇                                  ρ                  i                                                      +                                          D                1                            ⁢                                                ∇                  2                                ⁢                                  ρ                  i                                                                    ,                            (        2        )            with the concentration and diffusion coefficient for analyte i denoted by ρi and Di, respectively: u is the velocity vector. Equation (2) assumes there are no sources or sinks of ρi. A number of different analytes may be present in the sample, so the subscript i varies between 1 and the number of analytes. Species that obey equation (2) are known as conserved scalars because their transport can redistribute the species within the channel but not change the total amount present. Unless the species is present in concentrations high enough to affect the fluid viscosity μ, the species can be treated as a passive scalar: the velocity field can be determined independently of the species distributions, and then the species distributions can be solved after the velocity field is determined.
In many situations, such as in the exemplary embodiments discussed below, advection is assumed to be the dominant transport process and diffusive transport is therefore neglected. This assumption is reasonable for many micro-fluidic systems and applications, such as the exemplary embodiments discussed below, but is not necessary. In the situations where diffusive transport is neglected, the last term in equation (2) may be dropped and the advection alone may be feature of focus. Integrating along a streamline dx/dt=u gives
                              ρ          i                =                              constant            ⁢                                                  ⁢            along            ⁢                                          ⅆ                x                                            ⅆ                t                                              =                      u            .                                              (        3        )            
in other words, in the absence of diffusion, analytes flow along streamlines through the component.
In addition to describing the flow of a continuous variable such as the concentration of a species, the streamline dx/dt=u also gives the path of particles that travel at the local fluid velocity. A number of authors have used the transport of these “passive particles” to characterize the flow fields in micro-fluidic components, particularly mixers. A conventional approach includes releasing a number of particles upstream in the component, and examining the distributions of the particles in the cross-section at the component exit. This is the same concept as generating a Poincaré map to quantify the quality of mixing.
The following example focuses on an exemplary channel containing a single, isolated exemplary feature, and it demonstrates the characteristics of the flow that. This characteristic makes an aspect of the present invention effective, as discussed in further detail below. The channel has width w and height h, with w/h=3.11.
Consider channel 100 in FIGS. 1A and 1B with a feature that is a single diagonal groove 102 cut into channel floor 104. The width of groove 102 is w/4, the depth is 0.42h, and groove 102 runs across the channel at a 45° angle from sides 106 and 108. Solving the steady Stokes momentum equation∇p=/μ∇2u,  (4)subject to the mass conservation constraint for incompressible flow,∇·u=0,  (5)produces the velocity field u, where μ is the viscosity and p is the pressure. A zero-slip velocity boundary condition is assumed, so u=0 on all surfaces. The zero-slip assumption has been validated experimentally for the pressure-driven flow of water in channels as small as 40 nm. The computational grid extends far enough upstream and downstream from groove 102 to ensure that a fully developed velocity field approaches groove 102, and that the velocity returns to this fully developed profile after passing groove 102.
The flow through channel 100 was found by solving equations (2), (4) and (5) using a finite-volume incompressible flow solver for complex geometries, which includes an option to remove the inertial terms from its full Navier-Stokes solver in order to model Stokes flow. As illustrated in FIG. 1B, water enters channel 100 carrying two different dyes (light on the right 110 and dark on the left 112), and the three cross-sections 114, 116 and 118 show the distribution of these dyes at various points in channel 100. Streamline 120 in FIG. 1B follows the transport of light fluid originating at point A through to the downstream location at point B. As streamline 120 indicates, fluid that encounters upstream end 122 of the groove 102 goes into groove 102 and crosses to side 106. The result at exit plane 118 is a new dye distribution.
The dye distribution at exit plane 118 in FIG. 1B can be determined efficiently using a particle-tracking method. Passive particles of unknown color are distributed uniformly in exit plane 118. A streamline passing through each particle is traced upstream to the point where each particle entered the component, and that inflow location determines the color of each particle. FIG. 2 demonstrates this approach for the geometry in FIGS. 1A and 1B.
For Stokes flow, some flow properties, such as the pressure drop across the component, scale with flow rate, component size, and viscosity. The scalar distributions seen in planes 114, 116 and 118 in FIG. 1B are invariant—these are valid for any combination of geometry and flow conditions that satisfy the assumption of Stokes flow, given the inflow condition of two separate streams. Previous work demonstrated independence of the flow in mixing components below Re=10. For comparison, if w=1000 μm, water at 25° C. flowing at 0.1 mL min−1 through this geometry gives Re˜3.
FIG. 3 shows a projection of the entire streamline 120 in FIG. 1B onto exit plane 118 of the channel. The projection travels from point A, down into the groove, and out again, finally intersecting point B. The straight line going from A to B is the net effect of this transport. As was noted above, the inflow dye distribution and the coordinates of point A are sufficient to determine the fluid color at point B—the detailed shape of the streamline is not needed after the location of point A has been determined.
As shown by streamline 120 in FIG. 1B, the effects of groove 102 on the velocity field do not reach far upstream or downstream from groove 102. Streamline 120 follows the steady-state, parallel flow down channel 100 as it approaches groove 102 and then abruptly turns down into groove 102. After exiting groove 102, streamline 120 turns quickly to point directly down channel 100 parallel to the wall 106. This localized effect of groove 102 on the velocity field is a consequence of the insignificance of inertial terms in Stokes flow, and this effect has been observed for Navier-Stokes calculations at low Reynolds numbers. This effect is quantified in FIG. 4, which shows the deviation of the x velocity component from the steady-state distribution for channel 100 in the neighborhood of groove 102. Groove 102 extends from x/w=0 to x/w=1 in this plot. FIG. 4 shows root-mean-square values of the velocity deviation, defined as
                    RMS        =                                                            ∑                                  i                  =                  1                                ncells                            ⁢                                                (                                                                                    u                        i                                            -                                              U                        i                                                                                    u                      norm                                                        )                                2                                      ncells                                              (        6        )            
At each station along the channel, a grid of 64×22 cells was used in the velocity calculation, and in each cell, i, the steady-state x-velocity Ui (i.e., the velocity that cell would have if groove 102 were not there) is compared with the local x-velocity ui. This difference is squared and summed for all ncells=1408 in the cross-section to give a measure of how far the local distribution is from the steady-state distribution. The velocity unorm is used to normalize the deviation. Higher, solid curve 402 in FIG. 4 uses Ui in each cell to normalize the difference for that cell, and lower, dashed curve 404 uses the mean velocity in channel 100 to normalize the difference for all cells. Normalizing by Ui leads to a larger measure of the deviation because Ui in cells near the edges of channel 100 are very small; an increase in the velocity at these locations due to groove 102 leads to a large normalized difference.
In addition to the maximum deviation from steady-state, of interest is how far from groove 102 the effects of groove 102 reach. Approaching groove 102, the flow field stays within 1% of its steady-state velocity distribution as close as w/2 from upstream end 122 of groove 102. The flow then returns to within 1% of the steady-state distribution within a distance w/2 downstream from groove 102. Therefore groove 102 significantly alters the velocity field only in a very tight region immediately in front of and behind it. As the profile of fluid distribution in FIG. 1B demonstrate, however, once groove 102 redistributes the scalars in the fluid, this new distribution persists downstream of groove 102, frozen by the parallel steady-state flow.